#6524714

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499992
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#62824	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3036 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,6)(4,8)(5,7)(9,13)(10,12)(11,14)(15,16)(18,22,24,28)(19,25)(23,31,26,32)(27,29);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,18)(19,26)(20,28)(21,24)(22,30)(23,25)(27,32)(29,31);(2,7)(3,9)(4,11)(12,16)(17,19)(18,23)(20,29)(21,25)(22,31)(24,26)(27,30)(28,32);(1,4,5,11)(2,8,7,14)(3,10,9,15)(6,12,13,16)(19,25)(23,26)(27,29)(31,32);(3,9)(6,13)(10,15)(12,16)(18,24)(22,28)(23,26)(31,32)

Polycyclic presentation:

2^9|a^2=gh,d^2=hi,f^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6431910);